After our tour, you can take the Funicular down the hill. The ride is short, but scenic, or you can walk down the hill to Clark Ádám tér in about 10 minutes, if you prefer.

It'll be a late lunch, but I have two recommendations here. Going north (left, if you’re facing the river) about 180 m, you’ll find Ildikó Konyhája. If you want great Hungarian cuisine in a tiny, rustic atmosphere, this is the place to go. If you go south about the same distance, you’ll find Marvelosa Café. Come here for delicious Hungarian food in a fresh, whimsical setting. 

It will be well into the afternoon at this point and the rest of the day is yours. I have two suggestions for what you could do next, based on proximity. If you'd like to do them both, you can choose one for today and save the other for Wednesday, or do it all! The bath is open until 8:00; the Gellért Hill Cave church is open until 7:30! 

1. A short ride on Tram 19 (or a 20-minute walk) along the river will get you to Rudas Thermal Bath. I recommend the Turkish Bath ticket for about $30. The Turkish bath is incredible and you'll also be able to use the pools and saunas at the facility, as well as a locker and cabin (change room). To do it justice, expect to spend at least a few hours here. Tickets must be bought online and you need to bring a towel and sandals, as well as your swimwear. As it's Friday, you also have an option to buy a night ticket, which you can use from 10 pm to 3 am! More ticket information is available here.
2. Walk along this route. It will take you to Várkert Bazár (the garden entrance to the castle hill), the statue of Elizabeth Queen of Hungary, up Gellért Hill to the spectacular Liberty Monument. From there, you'll go back down the hill where you can find the entrance to the Gellért Hill Cave church. The entry fee is inexpensive and includes a thorough audio guide. 

If you are doing all of this, you may want to rearrange the walking route to visit the cave church first. This will ensure that you get there before it closes, and you may find yourself at the top of the hill at the Liberty Monument as the sun is setting to watch the city light up.